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Compactly supported tight frames associated with refinable functions. (English) Zbl 0948.42022
Summary: It is well known that in applied and computational mathematics, cardinal B-splines play an important role in geometric modeling (in computer aided geometric design), statistical data representation (or modeling), solution of differential equations (in numerical analysis), and so forth. More recently, in the development of wavelet analysis, cardinal B-splines also serve as a canonical example of scaling functions that generate multiresolution analyses of \(L^2(-\infty, \infty)\). However, although cardinal B-splines have compact support, their corresponding orthonormal wavelets (of Battle and Lemarié) have infinite duration. To preserve such properties as self-duality while requiring compact support, the notion of tight frames is probably the only replacement of that of orthonormal wavelets. In this paper, we study compactly supported tight frames \(\Psi= \{\psi^1,\dots, \psi^N\}\) for \(L^2(-\infty, \infty)\) that correspond to some refinable functions with compact support, give a precise existence criterion of \(\Psi\) in terms of an inequality condition on the Laurent polynomial symbols of the refinable functions, show that this condition is not always satisfied (implying the nonexistence of tight frames via the matrix extension approach), and give a constructive proof that when \(\Psi\) does exist, two functions with compact support are sufficient to constitute \(\Psi\), while three guarantee symmetry/anti-symmetry, when the given refinable function is symmetric.

MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A15 Spline approximation
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